Seems to me your math is confused (but correct in part).
The lens makes a 5mm DIAMETER perfect focus at a foot.
A different lense with a 2-foot focal length, can only make a perfectly focused 10mm image at 2 feet, not 25mm. The intensity of the light, anywhere in the image, goes down by 1/4 (area=diameter squared); or your lense needs 4 times the area to produce the same heat. And so on…
20-foot focal length 100mm sun image.
OTOH, the 1-foot focal as you move it out (or any other lense) the sun image becomes unfocussed and evern larger; the perfect focus is the smallest you can make the image of the sun.
So at 250 miles, the sun’s image will be 1/2 degree wide; or, about (250/114) if you know how to work in Radians, 1 radian is about 57 degrees, so half a degree is about 1/114 of a radian, and using sine law, sin(height/distance) is approx equal to (height/distance) for small angles.
(Basically, the image of a disc at effective infinite distance, but 30 minutes or 1/2 degree in diameter - with a spread of half a degree from the lens, lens focal distance 250 miles).
Twice as far away? The sun’s image is twice its previous diameter (assuming you are focusing so the diameter is at a minimum). If you keep the same F ratio then your lens/mirror also is twice the previous diameter.
So, the lens/mirror has collected 4 times a much light (area goes as diameter squared). But the spot/image of the sun also has 4 times the previous area (twice the size, four times the area again).
So, if you keep the F ratio constant, the intensity of the light (or power per amount of area) stays constant.
An I think the North Koreans have lauched their solar weapon and are now using it where I live. Its pushing 100 degrees and the heat index is hovering around 130right now. And I’m in the deep south and it sure as hell aint one of those nice dry heats either.
From the ant’s point of view, the Sun looks big. It occupies a larger angular area than without the magnifying glass. That is why the ant burns.
But the ant will never heat up more than the Sun. It will never get to a higher temp than the Sun, from radiation from the Sun, no matter what magnifying glass or mirrors you use.
They typically ARE but they don’t have to be. The important part to remember for this discussion is that whatever distance you optical system is from the target will be by definition the focal distance. That distance will determine the minimum size you can focus the sun down to. And once you have that, then that also determines how big your solar death ray machine will have to be achieve whatever “power level” you think you need.
This recent heat wave got me thinking. The OP was probably thinking along the lines of tanks exploding, buildings burst into flames, and people getting charr broiled in short order. And, I think we have shown that to do that your orbiting thingamabob is going to have to be absurdly big.
Now, if you are patient and don’t need all that drama you could probably get by with something much smaller. And used the savings to bedazzle your new lair.
Lets wait until you have a heat wave in some big city. Now, lets use our orbiting mirrors to say double the amount of sunlight hitting the area. If you could keep that up for a few hours I think you could heatstroke a buttload of people. And you ain’t evacuating a big city fast enough probably.
Or, even if your enemy comes up with a plan to evacuate the cities or build shelters or whatever you still have another option.
Hit agricultural areas. I suspect during a hot day there are plenty of crops/plants that if they were exposed to a modest increase in the amount of sunlight for not a terribly long period of time you’d kill em. Or for that matter an abnormal amount of heat/light during some other time of year might mess up their growth cycle.
The impact of being able to wipe out a few square miles of crops/plants every day would be an economic pain the ass at least for the target country even if nobody died or no infrastructure was damaged.
The big problem is, it would be pretty obvious what was happening and it would be an act of war. If you weren’t the USA, Russia or China, you’d be a friend of one of those. whoever is on your side would gladly demonstrate (with warheads) why the mirror in space violates far too many rules to be left alone, the primary one being no weapons in space, please.
So many square miles of active mirror would be a sitting duck. Just take out the command/control units and the mylar sheets would drift away aimlessly; I suspect with that surface area it would re-enter pretty soon, all the while being a tangled mess on unaimed garbage.
If there was a large enough sun focus in space, and it were big enough, then it would be terrible. Imagine if our earth was an ant, and the increase in intensity over the whole earth was by a factor of 100. This would be as if the Sun had become a red giant in the area in which the magnifying glass was focused. If that area was as large as the Earth, (and if it is actually possible to focus sunlight by a factor of 100 through an area as large as the Earth itself), then the oceans would boil, we would all die, and what is supposed to occur naturally through the evolution of the Sun through its red giant stage will be caused a billion years early.
Of course, the other planets will not be affected.
Such a weapon would only be workable, if not practical, if the country that used it used against a non-space capable country and the user country could also make it quite clear to any other space capable country that they best not be fucked with when doing their business elsewhere.
If you can get away with that, why ot just send a wave of bombers and cruise missiles in to blow the country into submission with shock and awe, followed by ground forces who expect to be greeted as liberators… that should get the mission accomplished and would be a lot cheaper than a billion dollars worth of mylar - wouldn’t it?
Actually, anyone capable of this tech has already agreed not to weaponize space, because that’s a really bad and expensive arms race toget into.